Assume that we have $m$ balls and $n$ bins. We throw the balls in bins randomly. Further, assume that $i<n$ and $0<\rho<1$.
What is the probability of having $j$ bins $B^1, B^2, \dots, B^j$ among the first $i$ bins such that $i\times \rho < j \leq i$ and $min(X^1, \dots, X^j)>k$, where $X^l$ denotes the number of balls in $B^l$?